From Euclidean Geometry to Knots and Nets

This document is the Accepted Manuscript of an article accepted for publication in Synthese. Under embargo until 19 September 2018. The final publication is available at Springer via https://doi.org/10.1007/s11229-017-1558-x.This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if (a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, (b) the information thus displayed is not metrical and (c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.Peer reviewe

Infinitesimal Idealization, Easy Road Nominalism, and Fractional Quantum Statistics

It has been recently debated whether there exists a so-called “easy road” to nominalism. In this essay, I attempt to fill a lacuna in the debate by making a connection with the literature on infinite and infinitesimal idealization in science through an example from mathematical physics that has been largely ignored by philosophers. Specifically, by appealing to John Norton’s distinction between idealization and approximation, I argue that the phenomena of fractional quantum statistics bears negatively on Mary Leng’s proposed path to easy road nominalism, thereby partially defending Mark Colyvan’s claim that there is no easy road to nominalism

How to think about informal proofs

This document is the Accepted Manuscript version of the following article: Brendan Larvor, ‘How to think about informal proofs’, Synthese, Vol. 187(2): 715-730, first published online 9 September 2011. The final publication is available at Springer via doi:10.1007/s11229-011-0007-5It is argued in this study that (i) progress in the philosophy of mathematical practice requires a general positive account of informal proof; (ii) the best candidate is to think of informal proofs as arguments that depend on their matter as well as their logical form; (iii) articulating the dependency of informal inferences on their content requires a redefinition of logic as the general study of inferential actions; (iv) it is a decisive advantage of this conception of logic that it accommodates the many mathematical proofs that include actions on objects other than propositions; (v) this conception of logic permits the articulation of project-sized tasks for the philosophy of mathematical practice, thereby supplying a partial characterisation of normal research in the fieldPeer reviewedFinal Accepted Versio

Alkali-calcic and alkaline post-orogenic (PO) granite magmatism: petrologic constraints and geodynamic settings

The end of an orogenic Wilson cycle corresponds to amalgamation of terranes into a Pangaea and is marked by widespread magmatism dominated by granitoids. The post-collision event starts with magmatic processes still influenced by subducted crustal materials. The dominantly calc-alkaline suites show a shift from normal to high-K to very high-K associations. Source regions are composed of depleted and later enriched orogenic subcontinental lithospheric mantle, affected by dehydration melting and generating more and more K- and LILE-rich magmas. In the vicinity of intra-crustal magma chambers, anatexis by incongruent melting of hydrous minerals may generate peraluminous granitoids bearing mafic enclaves. The post-collision event ends with emplacement of bimodal post-orogenic (PO) suites along transcurrent fault zones. Two suites are defined, (i) the alkali-calcic monzonite-monzogranite-syenogranite-alkali feldspar granite association characterised by [biotite + plagioclase] fractionation and moderate [LILE + HFSE] enrichments and (ii) the alkaline monzonite-syenite-alkali feldspar granite association characterised by [amphibole + alkali feldspar] fractionation and displaying two evolutionary trends, one peralkaline with sodic mafic mineralogy and higher enrichments in HFSE than in LILE, and the other aluminous biotite-bearing marked by HFSE depletion relative to LILE due to accessory mineral precipitation. Alkali-calcic and alkaline suites differ essentially in the amounts of water present within intra-crustal magma chambers, promoting crystallisation of various mineral assemblages. The ultimate enriched and not depleted mantle source is identical for the two PO suites. The more primitive LILE and HFSE-rich source rapidly replaces the older orogenic mantle source during lithosphere delamination and becomes progressively the thermal boundary layer of the new lithosphere. Present rock compositions are a mixture of major mantle contribution and various crustal components carried by F-rich aqueous fluids circulating within convective cells created around magma chambers. In favourable areas, PO suites pre-date a new orogenic Wilson cycle. (C) 1998 Elsevier Science B.V. All rights reserved

Les cortlandites et les roches associées de la région de Tin-Erit (Hoggar, Algérie)

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